Shortest distance for cuboid by taking a certain path in 3D

Question

I am stuck on part (b) of the question. Please see image above. How would I work out the shortest distance with reducing the amount of cheese the ant has to crawl through?

Answer 1

The Pythagorean theorem works in unlimited dimensions so, if she love cheese and would eat her way through, the distance $$d=\sqrt{20^2+20^2+30^2}=\sqrt{1700}\approx 41.23$$

If she hates cheese, the way along the edges is $d=20+20+30=70$

OTOH, if she walks half-way down a 20-side while walking horizontally the other full 20-side, she can the continue the rest of the way down the 20-side while walking the full length of the 30-side.

$$\sqrt{20^2+10^2}+\sqrt{10^2+30^2}=\sqrt{500}+\sqrt{1000}\approx 22.36 + 31.62\approx 53.98$$

Another way is to think of the 20X30 and a 20X20 sections as a single rectangle.

$$\sqrt{(30+20)^2+20^2}=\sqrt{2900}\approx 53.85$$